Properties

Label 26T27
Degree $26$
Order $3042$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:(C_3\times S_3)$

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Show commands: Magma

magma: G := TransitiveGroup(26, 27);
 

Group invariants

Abstract group:  $C_{13}^2:(C_3\times S_3)$
magma: IdentifyGroup(G);
 
Order:  $3042=2 \cdot 3^{2} \cdot 13^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,16,3,21,9,23)(2,25,6,22,5,26)(4,17,12,24,10,19)(7,18,8,14,11,15)(13,20)$, $(1,6,8)(2,9,4)(3,12,13)(7,11,10)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

39T39, 39T40

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{26}$ $1$ $1$ $0$ $()$
2A $2^{13}$ $39$ $2$ $13$ $( 1,26)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,23)(12,24)(13,25)$
3A1 $3^{4},1^{14}$ $26$ $3$ $8$ $(14,18,17)(15,21,26)(16,24,22)(19,20,23)$
3A-1 $3^{4},1^{14}$ $26$ $3$ $8$ $( 1, 4, 5)( 2,13, 8)( 3, 9,11)( 6,10, 7)$
3B1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(14,22,16)(15,18,19)(17,23,25)(20,24,21)$
3B-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(14,16,22)(15,19,18)(17,25,23)(20,21,24)$
3C $3^{8},1^{2}$ $338$ $3$ $16$ $( 1, 5, 4)( 2, 8,13)( 3,11, 9)( 6, 7,10)(14,15,24)(16,20,17)(18,25,23)(19,21,26)$
6A1 $6^{4},2$ $507$ $6$ $21$ $( 1,26)( 2,16,10,14, 4,22)( 3,19, 6,15, 7,18)( 5,25,11,17,13,23)( 8,21,12,20, 9,24)$
6A-1 $6^{4},2$ $507$ $6$ $21$ $( 1,26)( 2,22, 4,14,10,16)( 3,18, 7,15, 6,19)( 5,23,13,17,11,25)( 8,24, 9,20,12,21)$
13A1 $13,1^{13}$ $6$ $13$ $12$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$
13A-1 $13,1^{13}$ $6$ $13$ $12$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)$
13A2 $13,1^{13}$ $6$ $13$ $12$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$
13A-2 $13,1^{13}$ $6$ $13$ $12$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)$
13B1 $13^{2}$ $9$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13B-1 $13^{2}$ $9$ $13$ $24$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13B2 $13^{2}$ $9$ $13$ $24$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13B-2 $13^{2}$ $9$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
13C1 $13^{2}$ $18$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,25,23,21,19,17,15,26,24,22,20,18,16)$
13C2 $13^{2}$ $18$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13D1 $13^{2}$ $18$ $13$ $24$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13D-1 $13^{2}$ $18$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13D2 $13^{2}$ $18$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13D-2 $13^{2}$ $18$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
26A1 $26$ $117$ $26$ $25$ $( 1,14, 3,20, 5,26, 7,19, 9,25,11,18,13,24, 2,17, 4,23, 6,16, 8,22,10,15,12,21)$
26A-1 $26$ $117$ $26$ $25$ $( 1,16,13,26,12,23,11,20,10,17, 9,14, 8,24, 7,21, 6,18, 5,15, 4,25, 3,22, 2,19)$
26A5 $26$ $117$ $26$ $25$ $( 1,23, 9,21, 4,19,12,17, 7,15, 2,26,10,24, 5,22,13,20, 8,18, 3,16,11,14, 6,25)$
26A-5 $26$ $117$ $26$ $25$ $( 1,22, 4,18, 7,14,10,23,13,19, 3,15, 6,24, 9,20,12,16, 2,25, 5,21, 8,17,11,26)$
39A1 $13,3^{4},1$ $78$ $39$ $20$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,17)(15,21,26)(16,24,22)(19,20,23)$
39A-1 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,18,17)(15,21,26)(16,24,22)(19,20,23)$
39A2 $13,3^{4},1$ $78$ $39$ $20$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,17,18)(15,26,21)(16,22,24)(19,23,20)$
39A-2 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,18,17)(15,21,26)(16,24,22)(19,20,23)$
39A4 $13,3^{4},1$ $78$ $39$ $20$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,17,18)(15,26,21)(16,22,24)(19,23,20)$
39A-4 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,18)(15,26,21)(16,22,24)(19,23,20)$
39A8 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,26,17)(15,22,20)(16,18,23)(21,24,25)$
39A-8 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,26)(15,20,22)(16,23,18)(21,25,24)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Character table

35 x 35 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed